FORWARD MODELING

The traveltime for a ray that travels a distance d in an homogeneous medium with elliptical anisotropy and axis of symmetry forming an angle $\gamma$with respect to the vertical (Figure ) is

$$t \ =\ \sqrt{ \frac{(\Delta x \cos \gamma + \Delta z \sin \g... ... \cos \gamma + \Delta x \sin \gamma )^2} {V_{\parallel}^{2}} },$$ (1)

where $\sqrt{\Delta x^2 + \Delta z^2} \ =\ d$ and $V_{\parallel}$ and $V_{\perp}$ are the velocities in the directions parallel and perpendicular respectively to the axis of symmetry. In Appendix A, I explain how to derive this expression.

 

ray-and-axis
Figure 1 Ray traveling a distance d in a medium with tilted axis of symmetry. $\alpha$ and $\gamma$ are the angles of the ray and the axis of symmetry respectively with respect to the vertical.

If the model is described as a superposition of N homogeneous orthogonal regions, the traveltime for the i^th ray traveling across the j^th region is

$$t_{i,j} \ =\ \sqrt{ \frac{(\Delta x_{i,j} \cos \gamma_j + \... ...mma_j + \Delta x_{i,j} \sin \gamma_j)^2}{V_{\parallel_j}^2}. }$$ (2)

Note that each homogeneous region is characterized by three parameters: two velocities and the angle of the axis of symmetry with respect to the vertical. From now on I will refer to this parameters as interval parameters. In the previous equation, $\sqrt {\Delta x_{i,j}^2 + \Delta z_{i,j}^2}$ is the distance traveled by the i^th ray in the j^th cell. The sum of expressions like equation (2) can be used to compute the traveltime from source to receiver for a ray that travels in an heterogeneous media, assuming the ray path is known. Byun (1982) and Michelena (1992) explain how to do the ray tracing.

Figure  shows the type of model that I will consider in this paper. It consists of homogeneous elliptically anisotropic blocks separated by straight interfaces of variable dip (a_j) and intercept (b_j). I assume that all the axes of symmetry for the different layers lie in the same plane of the survey geometry. If $\Delta x_{i,j}$ and $\Delta z_{i,j}$ are defined as

$$\begin{eqnarray} \Delta x_{i,j} & = & x_{i,j+1} - x_{i,j} \\ \Delta z_{i,j} & = & z_{i,j+1} - z_{i,j} \end{eqnarray}$$ (1) (2)

the expression (2) for the traveltime t_i,j of the i^th ray in the j^th cell becomes

$$t_{i,j} \ =\ \sqrt{ {\Delta X_{i,j}}^2 S_{\perp_j}^2 + {\Delta Z_{i,j}}^2 S_{\parallel_j}},$$ (4)

where $S_{\perp_j}$, $S_{\parallel_j}$, $\Delta X_{i,j}$ and $\Delta Z_{i,j}$ are equal to

$$\begin{eqnarray} S_{\perp_j} & = & \frac{1}{V_{\perp_j}}, \nonumber \\ S_{\parallel_j} & = & \frac{1}{V_{\parallel_j}}, \nonumber\end{eqnarray}$$

$$\Delta X_{i,j} \ =\ \Delta x_{i,j} \cos \gamma_j + (a_{j+1} x_{i,j+1} + b_{j+1} - a_{j} x_{i,j} - b_{j}) \sin \gamma_j,$$

$$\Delta Z_{i,j} \ =\ - (a_{j+1} x_{i,j+1} + b_{j+1} - a_{j} x_{i,j} - b_{j}) \cos \gamma_j + \Delta x_{i,j} \sin \gamma_j.$$

(x_i,j, z_i,j) is the point of intersection between the i^th ray and the j^th interface. If the axis of symmetry is vertical ($\gamma_j = 0$), it follows that $\Delta X_{i,j} = \Delta x_{i,j}$ and $\Delta Z_{i,j} = \Delta z_{i,j}$.

 

inv-model
Figure 2 Model of velocities and heterogeneities. The top and bottom interfaces are horizontal (a₁ = a_N+1 = 0) and located at known depths.

Besides the interval parameters previously described, I have added two more parameters to describe how the boundaries that separate different intervals may change their positions. I call these parameters boundary parameters. Figure  shows how to count both intervals and boundaries.

The total traveltime for a ray that travels from source to receiver is

$$t_i( \mbox{\boldmath$m$}) \ =\ \sum_{j=1}^{N} t_{i,j} ( \mbox{\boldmath$m$}) \hspace{.5in} i=1,...,M,$$ (5)

where $\mbox{\boldmath$m$}$ is the vector of model parameters of 5N elements:

$$\begin{eqnarray} \mbox{\boldmath$m$} & = & (m_1,...,m_N,m_{N+1},...,m_{2N},m_{2N... ...,S_{\parallel_N}, \gamma_1,...,\gamma_N, b_1,...,b_N,a_1,...,a_N),\end{eqnarray}$$ (6)

and M is the total number of traveltimes.

Equation (5) is the system of nonlinear equations that relates the model parameters with the measured traveltimes. A linearized version of this equations will be used in the next section to solve the inverse problem. As explained in Figure , b₁ and a₁ are known. This makes the number of variables in $\mbox{\boldmath$m$}$ equal to 5N - 2.